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Journal of Mathematical Physics

On Doob h-transformations for finite-time quantum state reduction

Levent Ali Mengütürk

Published Mar 1, 2024• 2 citations
Publisher

Abstract

The paper develops a finite-time quantum state reduction framework via the use of Lévy random bridges (LRBs) that can be understood as Doob h-transformations on Lévy processes. Building upon the non-anticipative semimartingale representation of LRBs, we propose a family of energy-driven stochastic Schrödinger equations that go beyond the purely-continuous Brownian motion setup, and enter the scope of quantum systems involving discontinuities and heavy-tails. In doing so, we allow collapse dynamics to be governed intrinsically by the Markovian statistics of LRBs. The framework can host progressively convoluted stochastic state reduction dynamics in a tractable way and encapsulates jump-diffusion Schrödinger evolutions by use of the Lévy-Itô decomposition. Our construct motivates the usage of more general Markov processes and Doob h-transformations in producing consistent wave function collapse dynamics in finite-time.

DOI
10.1063/5.0162658

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The paper develops a finite-time quantum state reduction framework via the use of Lévy random bridges (LRBs) that can be understood as Doob h-transformations on Lévy processes. Building upon the non-anticipative semimartingale representation of LRBs, we propose a family of energy-driven stochastic Schrödinger equations that go beyond the purely-continuous Brownian motion setup, and enter the scope of quantum systems involving discontinuities and heavy-tails. In doing so, we allow collapse dynamics to be governed intrinsically by the Markovian statistics of LRBs. The framework can host progressively convoluted stochastic state reduction dynamics in a tractable way and encapsulates jump-diffusion Schrödinger evolutions by use of the Lévy-Itô decomposition. Our construct motivates the usage of more general Markov processes and Doob h-transformations in producing consistent wave function collapse dynamics in finite-time.
Abstract Recently there has been much progress in the development of stochastic models for state reduction in quantum mechanics. In such models, the collapse of the wave function is a physical process, governed by a nonlinear stochastic differential equation that generalizes the Schrödinger equation. The present paper considers energy-based stochastic extensions of the Schrödinger equation. Most of the work carried out hitherto in this area has been concerned with models where the process driving the stochastic dynamics of the quantum state is Brownian motion. Here, the Brownian framework is broadened to a wider class of models where the noise process is of the Lévy type, admitting stationary and independent increments. The properties of such models are different from those of Brownian reduction models. In particular, for Lévy models the decoherence rate depends on the overall scale of the energy. Thus, in Lévy reduction models, a macroscopic quantum system will spontaneously collapse to an eigenstate even if the energy level gaps are small.
This paper introduces a class of stochastic processes constructed by conditioning càdlàg processes to take a predetermined set of marginal laws at fixed points in time. We collectively refer to the elements of this class of processes as random n-bridges (RnBs), where n refers to the cardinality of the set of conditioning laws, which can be chosen arbitrarily. We prove that if the underlying càdlàg process is Markov, then its RnB is Markov for n = 1 and has a particular Markov-like property for n > 1. As a canonical subclass, we construct Brownian RnBs (BRnBs) and provide their anticipative representation as well as their non-anticipative semimartingale representation. This motivates the second part of this paper, wherein we apply BRnBs to describe an energy-based state reduction in a composite quantum system involving a set of commuting observables measured successively at different times. The highly tractable so-called energy-based quantum state reduction models are significant in their role as the most important alternative to the popular CLS model. We ask the random variables composing the anticipative form of a BRnB to take the eigenvalues of a set of compatible Hamiltonians and provide a new stochastic Schrödinger evolution on a complex Hilbert space that models sequential reduction dynamics of the eigenstates through a single wave function. By doing so, we also extend the Brody–Hughston finite-time collapse model to a many-body setup with sequential measurements.
A new energy-based stochastic extension of the Schrödinger equation for which the wave function collapses after the passage of a finite amount of time is proposed. An exact closed-form solution to the dynamical equation, valid for all finite-dimensional quantum systems, is presented and used to verify explicitly that reduction of the state vector to an energy eigenstate occurs. A time-change technique is introduced to construct a “clock” variable that relates the asymptotic and the finite-time collapse models by means of a nonlinear transformation.
A closed-form solution to the energy-based stochastic Schrodinger equation with a time-dependent coupling is obtained. Three different methods for establishing the result are shown: the first method uses the idea of conditioning on the trajectory of a random path; the second method is based on the change-of-measure technique in probability theory; and the third method relies on the decomposition of a random path into increments. The solution is given by a functional of a non-Markovian process; this functional process is shown nevertheless to possess the Markov property. When the coupling is time-independent, it is known that state reduction occurs asymptotically--that is to say, over an infinite time horizon. In the case of a time-dependent coupling, we show that if the magnitude of the coupling decreases sufficiently rapidly, then the energy variance will be reduced under the dynamics, but the state need not reach an energy eigenstate. We also construct an example where the opposite situation prevails, in which complete state reduction is achieved after the passage of a finite period of time.
A new energy-based stochastic extension of the Schrodinger equation for which the wave function collapses after the passage of a finite amount of time is proposed. An exact closed-form solution to the dynamical equation, valid for all finite-dimensional quantum systems, is presented and used to verify explicitly that reduction of the state vector to an energy eigenstate occurs. A time-change technique is introduced to construct a "clock" variable that relates the asymptotic and the finite-time collapse models by means of a nonlinear transformation. (c) 2005 American Institute of Physics
Starting from the forward and backward infinitesimal generators of bilateral, time-homogeneous Markov processes, the self-adjoint Hamiltonians of the generalized Schrödinger equations are first introduced by means of suitable Doob transformations. Then, by broadening with the aid of the Dirichlet forms, the results of the Nelson stochastic mechanics, we prove that it is possible to associate bilateral, and time-homogeneous Markov processes to the wave functions stationary solutions of our generalized Schrödinger equations. Particular attention is then paid to the special case of the Lévy-Schrödinger (LS) equations and to their associated Lévy-type Markov processes, and to a few examples of Cauchy background noise.